 Y of omega, the conformal element, also known in physics as energy momentum tensor, gives a series for which the coefficients are generators of berycel algebra with a certain fixed central charge. And there was a certain locality. There was a locality condition, y, which I'm not going to repeat. And then we also introduced a notion of modules. So this was a pair m, ym. M is again a complex vector space. And ym is a map similar to y. It's from v to endomorphisms m series, and there was also a locality condition on ym, which I didn't write. So I promised to give you some simple example, which will be relevant for the story about relation between vertex operator order percent for manifolds. But so far let me just give it as an example of just the UA without reference to for manifolds. So it's what in mass literature often called Heisenberg, the UA, so in physics, this is a mathematical structure which describes two-dimensional free. So in order to produce, so we want to start with h vector space, and it's equipped with symmetric, non-degenerate pairing, which is real valued. And then from this data, one can define Heisenberg, the algebra which already appeared yesterday in North stock, and it was also, it appeared in the context of Hilbert-Skinhoff points, which also will be relevant later for us, and also in order to fix some rotations, Heisenberg, the algebra. So let me denote it by Heisenberg of h. So as a vector space, it's the following. It's h tensor. It's a formal run power series in some form-viable t. So this can be understood as the loop space constructed from h, and I also add a certain central element, c. And so let me introduce the following notation. So if for h, for a, some element of my vector space h, I will denote, I can construct a corresponding element from this vector space a with index n, which by definition just a tensor with tn is an integer. Then the, so the algebra structure on this vector space is given by the following relation. For any pair of elements of h, the leap bracket of the corresponding elements here is equal to m times delta m minus m. The pairing of a and b times c is the central element. Okay. So now from this, so there one can guess, so before producing a vertex of the algebra, so one can introduce the notion of modules. Consider the following modules of this Heisenberg algebra, which also appeared, already appeared yesterday in a certain form. So let me denote by Fock. Let me fix some element lambda from my vector space h and denote Fock lambda of h is the highest rate module of my Heisenberg algebra, which is induced by the following representation of a sub-algebra generated by negative elements tensor, which is negative powers of t here. So it's induced by the following relation. So first, so what we denote by lambda is a highest weight or lowest, depends on how we want to highest weight vector, and it satisfies the following conditions. So a0 acting on lambda is equal to this, the pairing of a and lambda times lambda itself. So lambda is the eigenvector of any a0, for a0 corresponding to any element a from h, and an acting on lambda is 0 for any n greater, for any positive, strictly positive integer. And c acts the central element, acts by just by the identity. And of course, then some more explicitly, one can write this as a, so the Fock space, the lambda as vector space is generated by all possible, by the actions of all possible generators here with negative indices, where all mi is greater or equal to 0. So of course, you can also, I mean, it's also as a vector space, just as a morphic to a symmetric power of h tensor with powers series in t with negative powers. Any questions? So for many of you, this might be very standard thing, but as I learned sometimes, not for all mathematicians, this is actually, this is kind of standard for physicists. Okay, so, and then the statement that the Fock space, and I take this element to be 0, Fock 0 of h has structure vector operator algebra. And so let me introduce another notation. So for any element a in my vector space h, known by az, the formal powers, formal around powers in variable z, these coefficients being am, which were defined previously. So these are generators, these are elements of the Heisenbeck, the algebra. So then, so this defines me v as a vector space. So the z2 grading everywhere will be 0 here in this case. And that grading will be provided by the conformal element, which I will construct in a moment. So I need to define, to define a VOA structure, I need to define how this y, the y operator. So I need to define, so it's enough to define, of course, on any generator here. So this will be given by the following formula. Well, let me write it in this way. The product over all i from 1 to k, and then I take here, so in the product, I take the derivative. So for each guy here, I can start the function, the genetic function using the following rule, and I take its derivative with respect to that, the n i's derivative. And then I take the normal ordered product of this, which means, so this is also already up here, but let me remind you, so normal order means that, so we need to order all this guys, which are considered as elements of the Heisenberg algebra, according to the following rule. So a n with an index less than zero should be before a n's with positive x. So this gives me some series in Z, with coefficients being elements of the Heisenberg algebra, since this vector space is a representation of the Heisenberg algebra, this is an element, the coefficients are elements of the intermorphism of V as we want. And so I also need to say what is 1. So 1 here is just, the unit element is just the highest rate vector, and also I need to say what is omega, and omega is, well, so one way to define omega is to pick a basis, so let's E i to be a basis H, sorry, let's me with upper index, and E i with lower index would be dual basis, dual with respect to the pairing. And then this can be constructed as a following. So the exercise is to check that if I take y of this guy, the coefficients here will satisfy the other algebra relations. So here, let me, so this is a bit different from what you see. So yesterday there was, the conformal element was depending on the choice of the element here, but here it's kind of unique. So another notion which will be useful for us later is the notion of what is kind of vertex operators. So this is partly the source of the name of the vertex operator algebra. So one way to understand, to understand vector, vertex operators is as well. Okay, so, but before I do this, so here I can start the, the vector space of the, folk space with zero label of H, but of course by the similar construction, I can understand folk space with some arbitrary label, lambda of H is a model of VOA because lambda is any element of H. Lambda was an element of H itself. So to represent this, this folk model was constructing by, essentially this was defining, this lambda defined the, the weight of the high street vector. So it, so the, the corresponding Y, the Y for this, for this model is defined by, is defined by the same formula, right, because here this formula gives you some, some elements of the Heisenberg, the algebra, but of course since we already, we already constructed this as a model of Heisenberg, the algebra, defined by the same. Now the vertex operators can be understood as the following linear maps from the folk space. So again, let me choose, it will be labeled by lambda element of my vector space H, which I started with. And it can be understood as the following linear map from a folk space labeled by mu to a folk space labeled by mu plus lambda and this is a, censored with all around series in formal variable Z. Well, up to, up to some overall possibly real power of Z. The following power of Z. And so they're given by the following formula, by the following explicit formula, which we'll write in the moment. And so informally, you can write it as a normal order of the exponential of the following integral. So where, so again, this, you use a construction, you use a definition, which I wrote before, what this means. So this kind of gives you a series in Z with coefficient being lambda n's. But more explicitly, you can write it as follows. So this is Z to the power of lambda zero. There is some over negative numbers and times exponent, some over positive numbers times epsilon lambda, epsilon lambda is just operator, which takes the highest rate vector here to the highest rate vector here. So it commutes with all other elements of Heisenberg algebra except the element with index zero, which for which the commutation relation is fixed by this relation essentially. In particular, so one can consider the following quantity, which has a nice physical interpretation, but mathematically it defined as follows. So I start with the highest, say with a zero highest rate vector and act by the bunch of vertex operators depending on different variables. And then I consider its pairing with the element, which is dual highest rate vector. So using all the explicit formulas, one can compute it. This can be a little bit. And so of course it's only well defined because U lambda acts as follows in between different modules. It's only well defined when the sum of lambda i's is the same as mu. And in this case, one can choose this is given by the following. A nice formula, the product of i less than j is that i minus j to the power, which is the product of lambda i and lambda j. So here it uses the pairing in H, but I can define it just to be zero if this is not satisfied. Just by saying that the modules with different labels are not actually orthogonal. So physically, so this is something which will appear in the context of this multi-monopoly invariance, which we will talk about. But more generically, it's kind of in physics. Just let me make a following quick remark. So since this vertex operator algebra describes some two-dimensional quantum field theory in physics, this corresponds to a kind of a correlator on a sphere where you insert this vertex operator algebra at the points z1, z2, z3, and so on. But also at infinity, you insert, so this, the pairing up with this dual to high-speed vector probably by mu is equivalent to inserting u mu at infinity. But you can consider, just this is the definition of... Questions? Positive powers of z is... So here it's... Well, so here you have... I told you it's even out of effect. Well, here it's negative. So it's a positive power. But the point is that whatever you acted on the finite number of guys here survives. But here, infinite numbers. So it will be well-defined, actually. Because whatever you act on an element on this, always the finite number of these guys will contribute with negative powers. Well, the things which I wrote before, so this a minus n with n is greater than 0. They are creation. And those guys with positive n are annihilation. So that's why only... Since you kind of have a state with finite number of particles on these finite numbers, this guy will act non-trivial. You can kill on these finite number of guys. So the kind of second example, but in a sense it's an example which kind of built on this Heisenberg vector-superator algebra example. And it also will be relevant in the four-manifold story. So this is what is called lattice. And so let me fix lattice lambda in H. So what does it mean? So lambda is a subgroup of H. If I put H as a billion group. So it's a subgroup which is isomorphic to Z. So I said the free-a-billion group of the rank being the same as the dimension of H. And such that the pairing, if I take my pairing on H, which I had before, if I restrict this pairing to lambda, this will be valued in Z. And so by lambda star, I will denote a dual lattice. So this also can be understood as a subspace of H, such that kind of the maximal subspace of H, such as the pairing of elements from lambda star and lambda integer valid. So this is the lattice VoA associated to this lattice lambda. So again, as a vector space, it's really just the following direct sum of the folk modules of the Heisenberg Lie algebra, where we take the sum of labels, where labels belong to my lattice lambda, and then are... So now I need to define how Y acts here. So an arbitrary element of the space has the following form as we've seen before. So I need to tell you what... So this is the element of the folk space lambda. These ends are all non-negative, and this is given by the following formula. So first, I have the same product as before, factorial A, but I also want to... Apart from this, I also have inside the normal ordering, I have the vertex operator labelled by this lattice element lambda, while omega is the same. So remark... The similar remark from physics is described now, two-dimensional free... Well, maybe not... Well, let me say two-dimensional compact chiral valued. So before the boson was valued, and the target space is h, and now the target space is h divided by lambda. That's why it's compact. Now, we kind of extended... So before any folk space with any highest weight vector was a module, but now once we extended the vertex operator algebra to this, the kind of the modules... The possible modules should be more restricted. So the possible modules... I'll just follow some modules. This lattice vector operator algebra are, let me denote by m, mu, and so mu will be element of the dual lattice, but it will be... What will matter is only class of this element modula, the lattice itself. And it will be given by the following direct sum. When I take a sum of a lattice shifted by mu, and again, the action of the elements of the VLA, we would define just by the same formula, because we already know those guys as the modules of Heisenberg V-algebra. So, any questions about... So this was a crash course in vertex operator algebras. So now I want to construct... So the idea is to construct some vertex operator algebras associated to four manifolds. And so let me kind of first give some sort of brief detour... Well, not really detour, but so let me... What is the kind of physical construction of the vertex operator algebras associated to four manifolds? And then I kind of try to continue discussion, avoiding to this reference to the six-dimensional. So I take a six-dimensional 2,0 theory as labelled by some Lie algebra G, which is either... So it's a direct sum of a billion Lie algebras and simply laced Lie algebras. And then I consider... So this is some six-dimensional theory, then I consider topologically twisted computation for manifold M4. And this produces me some two-dimensional 0,2... Well, let's say quantum field theory, T... So I usually denote it by... Well, let me denote it with Tg M4. It will depend on four manifolds and the choice of my Lie algebra G here. So this is some weird physical... This contains too much information. So in general, the theory itself is not a topological environment of four manifolds. But what we want to take is... We want to consider the BPA spectrum of the theory. And the BPA spectrum of the theory should be a topological environment. And the BPA... It's known... The BPA spectrum of two-dimensional 0,2 series is described in terms of vertex operator algebras. So one, for example... Let me say the following words. So one way to understand this... So, for example, many of you are familiar that in 2D 2,2 theory, you can do a topological twist and there will be some sort of chiral algebra which describes the BPA spectrum of this... of Tukumotov's theory. And so here, instead of... kind of... a simple chiral algebra, you have a kind of richer structure. You have a vertex operator algebra structure. So, in general... So in general, here, we don't have any Lagrangian description. So, in principle, we don't know if there is any Lagrangian description here. But if there is a particular Lagrangian description here, as, for example, Gage linear signal model description or non-linear signal model description, there is a... So, from the data of the Lagrangian, there is a systematically systematic way to produce this vertex operator algebra. So, in particular, if the 2D theory is a sigma model with target, then this vertex operator algebra can be described as a global section of what is called shift of chiral differential operators. So, x here is a target. Since the target here is supposed to be a color model. And, for example, in certain examples, this construction indeed gives a certain sigma model, which actually related... In certain cases, the target x is related to the Higgs and Kuhl branches, which were talked about in the Kuhl-Kuhl-Kajima stock. Well, but you can always go to... If you want, you can go to an infrared fixed point of the sphere. Well, if you have some sort of... Just, for example, the 3-0.2 kind of chiral multiplet, then the resulting PoA is a beta-gamma system. But, in more general, if you have some sort of boson valid in this target S, you need to do this CDO construction. If you have some sort of GLSM descriptions, then you need to do some sort of this dream-filled circle of business to construct this. And, well, you kind of... The sigma model means... Well, if you wish, your pass integral is an integral over maps from a two-dimensional surface, which is a source, your word sheet. Obviously, you have a 2x. You can see the maps. You kind of integrate over maps. Well, they can be different. You can see the... Yes, but it can be... In general, it can have boundaries where you can put some boundary conditions, which are brains, deep brains. But this... Well, this kind of... The PPS, so the two-dimensional series there, quantum field series, they are not kind of still not well defined. Well, they are defined by kind of in the... In the right-fix point, by using this vertex-period algebra, you can define them mathematically. But you can also define even without going... Like, if you study some PPS part of the spectrum, there is usually kind of mathematically well-developed theory how to study the sigma models. There is no general relation, but I will consider some examples, kind of when... Well, not in this lecture, but later. Some examples. There is some correspondence between this X and your... For manifold, if you... For manifold of some special class, for example. So, for example, if you... For manifold is something like P1 times the Riemann surface somewhere, like root surface, then... So, you have some Riemann surface. So, for this Riemann surface, you can associate a kind of Higgs branch of the corresponding class S theory. Roughly speaking, this X here will be this Higgs branch of this class S theory. So, there is this type of... Yes. And it depends... Well, kind of... To be precise, it's a kind of smooth structure. Yes. I mean, so far, there is no general mathematical construction of this, but there are some cases of G and M4 where you can... Some sort of subclasses of four manifolds where you can do this construction, which I want to consider. And... Yeah. But kind of physics predicts... Like up to... Like if you assume that such object is well-defined, then this thing should also exist. Okay. So, what kind of general... Can one make some sort of general prediction about some general prediction of this vector operator algebra? So, as a vector space, kind of... The naive kind of prediction tells us that this should be the cohomology of the de-joint union of all naive in a certain sense. Which I will comment before. The de-joint union of instanton-modular spaces on my four-manifold M4 with some gauge group G, such that the... Some compact league group G, such that the algebra of G is small G. So, the ambiguity of side-choice of G I will not discuss right now. And... And... Well... Well, I mean, C2, like, literally... Literally, there is no definition of the term class for when G is not, like, SUN or EN, but there is a notion of characteristic class where you sum over. So, this sum... So, this corresponds to my Z-grading in DOI. And the Z2-grading which I mentioned is a homological-grading mod Z2-grading I denoted by F and Z-grading was denoted by L0. And... So, of course, this, for example, so this can fail if there is a contribution which is not mentioned. There may be some contribution from monopoles. No monopole from... So, in particular, if one takes a trace and this is... So, this is something which is kind of, in a sense, this can be predicted from physics, but it also was... So, for certain cases, it was... In particular, some work by Nkajima, it was... I mean, there was construction of the vertex-operate-algebra structure on such spaces. And... So, in particular, for example, if one takes a trace over this DOI of minus 1 to the F, Q to the L0, then this should be the sum, the generating sum for electric characteristic of this instant-on-model space. So, this is kind of one... So, but... Here, so one can construct a different... Starting from this VOA, it contains more structure than just the character. It contains more information. And this more information is just a vector space. It contains vertex... This vertex-algebra structure. Yes. Although we precise, you need to do some sort of virtual... I don't know. Well, as I said, this is kind of a naive... naive expectation. But, so let me kind of give you some examples of explicit constructions of vertex-operator-algebra associated to four manifolds. So, one example... So, for... When G is just one, in principle, one can do this for very general manifold, for manifold. And this will be... So, part of our goal... The goals will be to say that what is called multi-monopole environments, which I will introduce later, are related to, in a certain way, to these correlation functions in the vertex-operator-algebra. In the corresponding vertex-operator... So, I don't have much time left. So, let me... For simplicity, so let me... Let me consider... a... 12-smooth compact for manifold, but possibly it will have a boundary which I will denote by m3. You can imagine having something like this. And... So, let me, for simplicity, assume that it's simply connected. So, then... Then I can associate to the four manifold the VOA as follows. So, as a... So, as in Norse talk yesterday... So, let me write... Consider... Haisenberg Algebra. First, consider Haisenberg Algebra associated to the vector space h, which is the cohomology of from four visual coefficients. So, since it's simply connected, this contains only even degrees. So, let me write it like this. And let me consider lambda to be the second homology lattice of m4. And... So, this can be... By punctuality, this can be identified with relative cohomology in degree two. And the dual lattice... Well, the dual lattice is... can be identified with... So, this can be considered as a subspace in h2 m4 r. And the dual lattice, lambda, can be identified with cohomology lattice, which is also... a subspace here. So, the idea is to construct kind of a Haisenberg V-A Algebra for h0 and h4, and the lattice V-A Algebra for the corresponding lattice, lambda, inside of this subspace. So, more literally, more explicitly, this is the following direct sum of this fork, lambda, and... So, and indeed... So, using... So, this has a... has a full interpretation. So, using by Roginowski and Kojima construction, which again was already mentioned in the no stock, this can be understood as a... So, this can be... This is isomorphic. This is of modules of the Haisenberg Algebra constructed by H to the cohomology of the joint union of... So, okay, this is in the case when m4 is a complex projective surface. Okay, I'll finish in a moment where this m base torsion 3 of rank 1 torsion 3 shifts on s. And... So, here I don't have... I vary both the second and first. So, to be precise, this is a... So, I vary both the first chunk class and the second component of the chunk character. And... So, for any C1 this is isomorphic to the Hilberg scheme of G2 points. So, as was explained in no stock yesterday, there is an explicit construction of the Haisenberg... of the action of the Haisenberg Algebra constructed from this vector space on those... on those cahomologes. Okay, let me stop here.